11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 1 to ∞) 5 / (j² + 4)

Verified step by step guidance

1. Identify the given series: \( \sum_{j=1}^{\infty} \frac{5}{j^{2} + 4} \). We want to determine if this series converges or diverges.
2. Recognize that the terms \( \frac{5}{j^{2} + 4} \) are positive and decrease as \( j \) increases, since the denominator grows quadratically.
3. Compare the given series to a known benchmark series. Notice that \( \frac{5}{j^{2} + 4} < \frac{5}{j^{2}} \) for all \( j \geq 1 \). The series \( \sum_{j=1}^{\infty} \frac{5}{j^{2}} \) is a constant multiple of the p-series \( \sum \frac{1}{j^{2}} \) with \( p = 2 > 1 \), which is known to converge.
4. Apply the Comparison Test: since \( \sum \frac{5}{j^{2}} \) converges and \( \frac{5}{j^{2} + 4} \leq \frac{5}{j^{2}} \), the original series \( \sum \frac{5}{j^{2} + 4} \) also converges.
5. Conclude that the series converges by the Comparison Test, justifying the answer based on the behavior of the terms and the known convergence of the p-series.

Key Concepts

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.

Comparison Test

The Comparison Test involves comparing a given series to a second series whose convergence behavior is known. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges.

p-Series Test

A p-series is of the form ∑ 1/n^p, which converges if p > 1 and diverges otherwise. Recognizing that the given series resembles a p-series helps in applying this test to determine convergence by comparing the denominator's growth rate.